Evaluation of \(\Phi _{\mathrm {MCMC}}\)  requires only the
mean and covariance from the GP emulator. Furthermore \(\Phi _{\mathrm {MCMC}}\)  is smooth and so is suitable for use within an MCMC algorithm to generate samples from the approximate posterior distribution of the parameters. [1]} contains further discussion of MCMC using GPs to emulate
the forward model, including situations where
data comes from finite time-averages but the emulator
is designed to approximate the infinite time-horizon forward model.
